L(s) = 1 | − 2-s − 3-s + 4-s + 2.23·5-s + 6-s − 8-s − 2·9-s − 2.23·10-s + 4.47·11-s − 12-s − 6.47·13-s − 2.23·15-s + 16-s − 3·17-s + 2·18-s + 4·19-s + 2.23·20-s − 4.47·22-s − 0.472·23-s + 24-s + 6.47·26-s + 5·27-s + 6.23·29-s + 2.23·30-s − 1.76·31-s − 32-s − 4.47·33-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.999·5-s + 0.408·6-s − 0.353·8-s − 0.666·9-s − 0.707·10-s + 1.34·11-s − 0.288·12-s − 1.79·13-s − 0.577·15-s + 0.250·16-s − 0.727·17-s + 0.471·18-s + 0.917·19-s + 0.499·20-s − 0.953·22-s − 0.0984·23-s + 0.204·24-s + 1.26·26-s + 0.962·27-s + 1.15·29-s + 0.408·30-s − 0.316·31-s − 0.176·32-s − 0.778·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
good | 3 | \( 1 + T + 3T^{2} \) |
| 5 | \( 1 - 2.23T + 5T^{2} \) |
| 11 | \( 1 - 4.47T + 11T^{2} \) |
| 13 | \( 1 + 6.47T + 13T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 0.472T + 23T^{2} \) |
| 29 | \( 1 - 6.23T + 29T^{2} \) |
| 31 | \( 1 + 1.76T + 31T^{2} \) |
| 37 | \( 1 + 8.47T + 37T^{2} \) |
| 43 | \( 1 - 9T + 43T^{2} \) |
| 47 | \( 1 + 4.94T + 47T^{2} \) |
| 53 | \( 1 + 6.70T + 53T^{2} \) |
| 59 | \( 1 + 3.52T + 59T^{2} \) |
| 61 | \( 1 + 1.29T + 61T^{2} \) |
| 67 | \( 1 + 11.4T + 67T^{2} \) |
| 71 | \( 1 - 2.23T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 2.70T + 79T^{2} \) |
| 83 | \( 1 - 2.94T + 83T^{2} \) |
| 89 | \( 1 - 3T + 89T^{2} \) |
| 97 | \( 1 + T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.144266088889297967355070265649, −7.21701023445518339067007334277, −6.63605297134577361262915198317, −5.98061143500565850461054530073, −5.25760794332877520231020417073, −4.47619137220741030084960298606, −3.11077197980668492156709920257, −2.28815566204990683242665584370, −1.33183159011323951243990733491, 0,
1.33183159011323951243990733491, 2.28815566204990683242665584370, 3.11077197980668492156709920257, 4.47619137220741030084960298606, 5.25760794332877520231020417073, 5.98061143500565850461054530073, 6.63605297134577361262915198317, 7.21701023445518339067007334277, 8.144266088889297967355070265649